Algebra Sleuth: Proof that 1 = 2?
Challenge your high school student to find the flaw in this short mathematical proof that one is equal to two. This activity provides a good review of basic math principals and the structure of mathematical proofs. It’s also a good reminder that knowing math principals is good protection against getting tricked.
What You Need:
- Proof that 1 = 2 (see below)
What You Do:
- Show your teen the proof.
- Ask her to tell you which step is invalid. She should determine both which number is wrong, and why.
- Help her keep going until she understands the answer.
The Proof that 2 = 1
1) a = b 1) Given
2) a2 = ab 2) Multiply both sides by a
3) a2-b2 = ab-b2 3) Subtract b2 from both sides
4) (a+b)(a-b) = b(a-b) 4) Factor both sides
5) (a+b) = b 5) Divide both sides by (a-b)
6) a+a = a 6) Substitute a for b
7) 2a = a 7) Addition
8) 2 = 1 8) Divide both sides by a
Part one: Step five is wrong. The rules of mathematics do not allow us to divide by zero.
Since a and b are equal, (a-b) = 0. Therefore, we cannot divide by (a-b)!
Note: To explain why you can't divide something by zero, ask your student how she would divide a pizza into 0 pieces. Impossible! The fewest number of pieces she could make would be one piece—the whole pizza!